AQFT and operator algebra
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The Gelfand spectrum (originally Гельфанд) of a commutative C*-algebra $A$ is a topological space $X$ such that $A$ is the algebra of complex-valued continuous functions on $X$. (This “Gelfand duality” is a special case of the general duality between spaces and their algebras of functions.)
Given a unital, not necessarily commutative, complex C*-algebra $A$, the set of its characters, that is: continuous nonzero linear homomorphisms into the field of complex numbers, is canonically equipped with what is called the spectral topology which is compact Hausdorff. (If applied to a nonunital $C^*$-algebra, then it is only locally compact.) This correspondence extends to a functor, called the Gel’fand spectrum from the category C*Alg of unital $C^*$-algebras to the category of Hausdorff topological spaces.
A character on a unital Banach algebra is automatically a continuous function (with Lipschitz constant 1).
The Gelfand spectrum functor is a full and faithful functor when restricted to the subcategory of commutative unital $C^*$-algebras.
The kernel of a character is clearly a codimension-$1$ closed subspace, and in particular a closed maximal ideal in $A$; therefore the Gel’fand spectrum is a topologised analogue of the maximal spectrum of a discrete algebra.
For noncommutative $C^*$-algebras the spaces of equivalence classes of irreducible representations (i.e., the spectrum) and their kernels (i.e., the primitive ideal space) are more important than the character space.
The Gelfand spectrum is also useful in the context of more general commutative Banach algebras.